The Propagators for δ and δ′ Potentials With Time-Dependent Strengths

Abstract

We study the time-dependent Schrödinger equation with finite number of Dirac δ and δ ′ potentials with time dependent strengths in one dimension. We obtain the formal solution for generic time dependent strengths and then we study the particular cases for single delta potential and limiting cases for finitely many delta potentials. Finally, we investigate the solution of time dependent Schrödinger equation for δ ′ potential with particular forms of the strengths.

Schrödinger evolution of superoscillations with $$\delta $$- and $$\delta '$$-potentials

“…By comparing (33a) with (30) we see that, besides the presence of the prefactor sgn(x), both coincide if α is replaced by β. Hence the same computations as in then proof of Lemma 4.2 lead to operators U (0) δ ′ ,± (t, x) and corresponding constants S (0) B,δ ′ ,± (t, x) for ϕ (0) δ ′ (t, ±x; ±k).…”

“…A detailed discussion of Schrödinger operators with δ and δ ′ -point interactions can be found in the standard monograph [13]. For further reading on Schrödinger operators with singular potentials we refer the reader to, e.g., [16,17,24,25,26,29,30,31,32,33,38,40,41,42,43,44] and the references therein.…”

“…For further reading on Schrödinger operators with singular potentials we refer the reader to, e.g., [16,17,24,25,26,29,30,31,32,33,38,40,41,42,43,44] and the references therein.…”